The page you're referring to is about the process for employers, rather than employees.
From an employer's perspective, as long as the money they paid out was for qualifying costs up to 8k, they don't have to report anything to HMRC. But most employers would want to be able to prove that it was for qualifying costs in case they got audited.
So this is really a question between you and your employer. You should talk to them as soon as possible to ask if they will pay qualifying costs tax-free, and ask them what evidence they will want to enable them to do that. And in the meantime make sure you have a clear record of those costs and keep receipts.
Ideally they would be happy to pay anything qualifying tax-free and the rest of the payment as a taxable benefit. But it may be that they don't actually want the administrative hassle, in which case you could be out of luck.